(1.8) The Poisson Process

❮ Poisson means fish in French ❯ ~ Molly McGuinness (2023)

Just another way to model something that happens in the real world.

This time, instead of modelling a coin toss in order to understand the behaviour of the outcome: heads or tails.

We model some stochastic process where events occur randomly in time, like customers arriving at a store. By modelling it as a Poisson process, we unlock the ability to know exactly about:

• The number of events that will happen in a window of time ~ [Poisson]
• The time we will have to wait to see the first or next event ~ [Exponential]
• The time we will have to wait to see the nth event ~ [Gamma]

By making some assumptions about this process, we can model it in an extremely useful way.

This gives birth to some distributions which describe different aspects of this process.

Simply put; it’s anything where events occur at some average rate.

What is a Poisson Process? §

Stop! ⚠️ Before we go further, we must understand the Poisson Process well - it’s super important, as many of the following distributions all describe different parts of a poisson process - not just the Poisson distribution!

A Poisson process is a type of stochastic process where events occur randomly in time. It is characterized by the following properties:

1. The number of events in non-overlapping intervals is independent.
2. The probability of more than one event happening in an infinitesimally small interval is essentially zero.
3. The probability of exactly one event in the infinitesimally small time interval ( dt ) is approximately $\lambda dt$, where $\lambda$ is the rate of the process.

Some examples of Poisson Processes

Category	Example
Physics	The number of particles emitted from a radioactive substance within a specific time frame.
Customer Service	The number of incoming calls to a customer service center within an hour.
Web Analytics	The number of hits or visits a website receives within a certain period.
Transportation	The number of cars passing a specific point on a highway in a given time frame.
Banking	The number of customers arriving at a bank teller or ATM in a certain period.
Public Transport	The number of buses arriving at a particular bus stop during a set time interval (assuming buses come irregularly).
Geology	The number of earthquakes in a specific region over a year.
Biology	The number of mutations in a particular stretch of DNA after certain exposure.
E-commerce	The number of orders received on an e-commerce platform in a day.
Social Media	The number of posts or tweets containing a specific hashtag within a set time frame.

---

Q6) Label with the distribution responsible for each part of the Poisson Process §

---

QX) Fill in the blanks §

Distribution	Continuous Analog
Geometric Distribution	_________
Negative Binomial	_________
Binomial Distribution	_________ (Related)
Poisson Distribution	_________ (Related)

Answer: “Exponential Distribution”, “Gamma Distribution”, and “Beta Distribution” x2